Problem: The lifespans of turtles in a particular zoo are normally distributed. The average turtle lives $96$ years; the standard deviation is $16.4$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a turtle living between $112.4$ and $128.8$ years.
Answer: $96$ $79.6$ $112.4$ $63.2$ $128.8$ $46.8$ $145.2$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $96$ years. We know the standard deviation is $16.4$ years, so one standard deviation below the mean is $79.6$ years and one standard deviation above the mean is $112.4$ years. Two standard deviations below the mean is $63.2$ years and two standard deviations above the mean is $128.8$ years. Three standard deviations below the mean is $46.8$ years and three standard deviations above the mean is $145.2$ years. We are interested in the probability of a turtle living between $112.4$ and $128.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the turtles will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the turtles will have lifespans within 1 standard deviation of the mean. The probability of a particular turtle living between $112.4$ and $128.8$ years is $\color{orange}{13.5\%}$.